Now we aim to control the time derivatives of the normalized density and pressure. To do this, we need a semi-convexity estimate in a similar vein to the Aronson-Bénilan estimate. The famous Aronson-Bénilan (see [1]} for a proof) estimate says that for the case of \(m(x,t) \equiv 1, f = 0\) , and \(\vec{b} = \vec{0}\)  that
\( \Delta p_k \ge -\frac{1}{(k-1)t}. \) 
